I realized several important things in my freshmen year at college:
- That my real talent lies in academics and not in organizing;
- That my real interest lies in economics and not in literature and philosophy;
- That I frankly am not good at mathematics.
The latest is not a life-changing discovery by any means, certainly not comparable to the first two; but it is important. To elaborate, I don’t think anybody in fact is really good at mathematics. One begins to realize quickly after studying point-set topology that pure mathematics is far less about the quick wits and clever intuition that underlies success in high school algebra, and far more about hard work – a willingness to spend hours and hours on what seems to be a trivial problem, but which you simply cannot grasp.
Even the old trick of staying up until mornings to solve the problem no longer works; some concepts crack only at the right moment, often when you least expect it. I had always thought that Einstein’s famous quote (of genius being 90% perspiration) and more recently Terence Tao’s comments were merely modesty. Now I understand that there is, of course, no modesty involved when one consider the kind of work ethic that it takes to be such successful academics.
But hard work is not impossible. I have found that there is a far more substantial limit to my mathematical ability, namely the state of my mind. I trust that the passage that follows will be a sufficient explanation for what I mean here. From the great French mathematician Alexander Grothendieck:
“…Since then I’ve had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group, who were much more brilliant, much more “gifted” than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle — while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end.Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects. In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of 30 or 35 years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birth-right, as it was mine: the capacity to be alone.”